Structure of shape derivatives around irregular domains and applications

Jimmy Lamboley (1) , Michel Pierre (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

In this paper, we describe the structure of shape derivatives around sets which are only assumed to be of finite perimeter in $\R^N$. This structure allows us to define a useful notion of positivity of the shape derivative and we show it implies its continuity with respect to the uniform norm when the boundary is Lipschitz (this restriction is essentially optimal). We apply this idea to various cases including the perimeter-type functionals for convex and pseudo-convex shapes or the Dirichlet energy of an open set.

Mots clés: Shape optimization; shape derivatives; sets of finite perimeter; convex sets; Dirichlet energy

Domaines: Optimisation et contrôle [math.OC]

Jimmy Lamboley, Michel Pierre. Structure of shape derivatives around irregular domains and applications. Journal of Convex Analysis, 2007, 14 (4), pp.807-822. ⟨hal-00095949⟩